On the Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency
JJ. Lange, A. Mavridou, L. Safina, A. Scalas (Eds.):13th Interaction and Concurrency Experience (ICE 2020).EPTCS 324, 2020, pp. 78–95, doi:10.4204/EPTCS.324.7
On the Parameterized Complexity of Synthesizing BooleanPetri Nets With Restricted Dependency
Ronny Tredup
Universit¨at Rostock, Institut f¨ur Informatik, Theoretische InformatikAlbert-Einstein-Straße 22, 18059, Rostock, GErmany [email protected]
Evgeny Erofeev
Department of Computing Science, Carl von Ossietzky Universit¨at Oldenburg,D-26111 Oldenburg, Germany [email protected]
Modeling of real-world systems with Petri nets allows to benefit from their generic concepts ofparallelism, synchronisation and conflict, and obtain a concise yet expressive system representation.Algorithms for synthesis of a net from a sequential specification enable the well-developed theoryof Petri nets to be applied for the system analysis through a net model. The problem of τ -synthesisconsists in deciding whether a given directed labeled graph A is isomorphic to the reachability graphof a Boolean Petri net N of type τ . In case of a positive decision, N should be constructed. Formany Boolean types of nets, the problem is NP-complete. This paper deals with a special variantof τ -synthesis that imposes restrictions for the target net N : we investigate dependency d-restricted τ -synthesis (DR τ S) where each place of N can influence and be influenced by at most d transitions.For a type τ , if τ -synthesis is NP-complete then DR τ S is also NP-complete. In this paper, we showthat DR τ S parameterized by d is in XP. Furthermore, we prove that it is W [ ] -hard, for many Booleantypes that allow unconditional interactions set and reset . Petri nets are widely used for modeling of parallel processes and distributed systems due to their abilityto express the relations of causal dependency, conflict and concurrency between system actions. Insystem analysis, one aims to check behavioral properties of such models, and many of these propertiesare decidable [12] for Petri nets and their reachability graphs which represent systems’ behaviors. Thetask of system synthesis is opposite: A system model has to be constructed from a given specificationof the desired behavior. Labeled transition systems serve as a convenient formalism for the behavioralspecification, and the goal is then to construct a Petri net whose reachability graph is isomorphic to theinput transition system. The relevance of the interest to the synthesis is justified in several ways. Incomparison to the sequential description of the system given by a transition system, the presence ofconcurrency/parallelism in a Petri net on a fine-grained level allows to encompass the full interleavingin the behavior in a concise yet clear way. As a result, this yields a usually much more compact systemmodel without loss of the expressiveness, as long as the synthesis terminates successfully. Besides, thealorithms of automatic synthesis ensure that the constructed model is correct-by-design, and hence itdoes not require any further verification. Moreover, the well-developed theory of Petri nets [12, 13]suggests a wide range of methods and techniques for behavioral and structural analysis of the synthesisedmodel, supporting possible improvements and optimization purposes in the initial system. Altogether, onny Tredup and Evgeny Erofeev true if the place is marked and false otherwise. In a Boolean Petri net, aplace p and a transition t are related by one of the Boolean interactions : no operation ( nop ), input ( inp ), output ( out ), unconditionally set to true ( set ), unconditionally reset to false ( res ), inverting ( swap ), test iftrue ( used ), and test if false ( free ). These interactions define in which way p and t influence each other:The interaction inp ( out ) defines that p must be true ( false ) before and false ( true ) after t ’s firing; free ( used ) implies that t ’s firing proves that p is false ( true ); nop means that p and t do not affect each otherat all; res ( set ) implies that p may initially be both false or true but after t ’s firing it is false ( true ); swap means that t inverts p ’s current Boolean value.Boolean Petri nets are classified by the sets of interactions between places and transitions that can beapplied. A set τ of Boolean interactions is called a type of net . A net N is of type τ (a τ -net ) if it appliesat most the interactions of τ . For a type τ , the τ - synthesis problem consists in deciding whether a given transition system A is isomorphic to the reachability graph of some τ -net N , and in constructing N if itexists. The complexity of synthesis strongly depends on the target Boolean type of nets. Thus, while τ -synthesis for elementary net systems (the case of τ = { nop , inp , out } ) is shown to be NP-complete [2],the same problem for flip-flop nets ( τ = { nop , inp , out , swap } ) is polynomial [17].This paper addresses the computational complexity of a special instance of τ -synthesis called De-pendency d-Restricted τ -Synthesis (DR τ S), which sets a limitation for the number of connections of aplace. This synthesis setting targets to those τ -nets in which every place must be in relation nop with allbut at most d transitions of the net, while the synthesis input is not confined. In system modeling [15],places of Petri nets are usually meant as conditions or resources, while transitions are meant as actions oragents. Hence, the formulation of d -restricted synthesis takes into consideration not only the concurrencyperspective but also possible a priori limitations on the number of agents which compete for the accessto some resource in the modeled system. From the theoretical perspective, the problem of synthesishas been extensively studied in the literature for the conventional Petri nets and their subclasses, whichare often defined via various structural restrictions: Recently, improvements of the existing synthesistechniques have been suggested for choice-free (transitions cannot share incoming places) [7], weightedmarked graphs (each place has at most one input and one output transition) [10, 11], fork-attribution(choice-free and at most one input for each transition) [27] and other net classes [6, 26]. In these works,the limitations were mainly subject to the quantity of connections between places and transitions. Onthe other hand, the results on synthesis of k -bounded (never more than k tokens on a place) [20], safe(1-bounded) and elementary nets [3] investigate classes which are defined through behavioral restrictions.Further, generalized settings of the synthesis problem for these and some other classes were studied [21],and NP-completeness results for many of them were presented. In contrast to this multitude of P/T netclasses, for Boolean nets, only the constrains for the set of interactions have appeared in the literature,deriving for instance flip-flop nets [17], trace nets [5], inhibitor nets [14]. This kind of constrain can beconsidered as behavioral limitation, leaving out the question of synthesis of possible structurally definedsubclasses of Boolean nets. The present paper aims to piece out the shortage by investigating the notionof d -restriction which limits the amount of connections between a place and transitions. The notion was0 Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency initially introduced in [22], where the complexity of d -restricted synthesis has been studied for a numberof Boolean types, and the W[1]-hardness of this problem has been proven. The current paper extendsthe previous work and tackles the problem for many types that necessarily include interactions res and set . We demonstrate the W[2]-hardness of d -restricted synthesis for these types, which makes a cleardistinction to the earlier results.The paper is organized as follows. After introducing of the necessary definitions in Section 2, themain contributions on W[2]-hardness of DR τ S are presented in Section 3. Section 4 suggests an outlookof the further research directions. Due to space restrictions, we omit some proofs, which can all be foundin the technical report [23].
In this section, we introduce the notions used throughout the paper and support them by examples.
Parameterized Complexity.
Due to space restrictions, we only give the basic notions of Parame-terized complexity (used in this paper) and refer to [9] for further related definitions. A parameterized problem is a language L ⊆ Σ ∗ × N , where Σ is a fixed alphabet and N is the set of natural numbers. For aninput ( x , k ) ∈ Σ ∗ × N , k is called the parameter . We define the size of an instance ( x , k ) , denoted by | ( x , k ) | ,as | x | + k , that is, k is encoded in unary. Let f , g : N → N be two computable functions. The parameterizedlanguage L is slice-wise polynomial (XP), if there exists an algorithm A such that, for all ( x , k ) ∈ Σ ∗ × N ,algorithm A decides whether ( x , k ) ∈ L in time bounded by f ( k ) · | ( x , k ) | g ( k ) ; if the runtime of A is evenbounded by f ( k ) · | ( x , k ) | O ( ) , then L is called fixed-parameter tractable (FPT). In order to classify param-eterized problems as being FPT or not, the W-hierarchy FPT ⊆ W [ ] ⊆ W [ ] ⊆ · · · ⊆ XP is defined [9,p. 435]. It is believed that all the sub-relations in this sequence are strict and that a problem is not FPTif it is W [ i ] -hard for some i ≥
1. Let L , L ⊆ Σ ∗ × N be two parameterized problems. A parameterized reduction from L to L is an algorithm that given an instance ( x , k ) of L , outputs an instance ( x (cid:48) , k (cid:48) ) of L in time f ( k ) · | x | O ( ) for some computable function f such that ( x , k ) is a yes-instance of L if and onlyif ( x (cid:48) , k (cid:48) ) is a yes-instance of L and k (cid:48) ≤ g ( k ) for some computable function g . If L is W [ i ] -hard andthere is a parameterized reduction from L to L , then L is W [ i ] -hard, too. Transition Systems.
A (deterministic) transition system (TS, for short) A = ( S , E , δ ) is a directedlabeled graph with the set of nodes S (called states ), the set of labels E (called events ) and partial transitionfunction δ : S × E −→ S . If δ ( s , e ) is defined, we say that event e occurs at state s , denoted by s e . An initialized TS A = ( S , E , δ , ι ) is a TS with a distinct initial state ι ∈ S where every state s ∈ S is reachable from ι by a directed labeled path. Boolean Types of Nets [3].
The following notion of Boolean types of nets allows to capture all
Boolean Petri nets in a uniform way. A
Boolean type of net τ = ( { , } , E τ , δ τ ) is a TS such that E τ is asubset of the Boolean interactions : E τ ⊆ I = { nop , inp , out , set , res , swap , used , free } . Each interaction i ∈ I is a binary partial function i : { , } → { , } as defined in Figure 1. For all x ∈ { , } and all i ∈ E τ ,the transition function of τ is defined by δ τ ( x , i ) = i ( x ) . Since a type τ is completely determined by E τ ,we often identify τ with E τ . τ -Nets. Let τ ⊆ I . A Boolean Petri net N = ( P , T , f , M ) of type τ (a τ -net ) is given by finite disjointsets P of places and T of transitions , a (total) flow function f : P × T → τ , and an initial markingM : P −→ { , } . A transition t ∈ T can fire in a marking M : P −→ { , } if δ τ ( M ( p ) , f ( p , t )) is definedfor all p ∈ P . By firing, t produces the marking M (cid:48) : P −→ { , } where M (cid:48) ( p ) = δ τ ( M ( p ) , f ( p , t )) forall p ∈ P , denoted by M t M (cid:48) . The behavior of τ -net N is captured by a transition system A N , called the reachability graph of N . The states set RS ( N ) of A N consists of all markings that can be reached from onny Tredup and Evgeny Erofeev M by sequences of transition firings. The dependency number d p = |{ t ∈ T | f ( p , t ) (cid:54) = nop }| of a place p of N is the number of transitions whose firing can possibly influence p or be influenced by themarking of p . The dependency number d N of a τ -net N is defined as d N = max { d p | p ∈ P } . For d ∈ N , a τ -net is called (dependency) d-restricted if d N ≤ d . Example 1.
Figure 2 shows the type τ = { nop , inp , swap } and the -restricted τ -netN = ( { R , R } , { a , b } , f , M ) with places R , R , flow-function f ( R , a ) = f ( R , b ) = inp , f ( R , b ) = nop ,f ( R , a ) = swap and initial marking M = ( M ( R ) , M ( R )) = ( , ) . Since inp ∈ τ and swap ∈ τ ,the transition a can fire in M , which leads to the marking M = ( M ( R ) , M ( R )) = ( , ) . After that, bcan fire, which results in the marking M (cid:48) = ( M (cid:48) ( R ) , M (cid:48) ( R )) = ( , ) . The reachability graph A N of N isdepicted on the right hand side of Figure 2. τ -Regions. Let τ ⊆ I . If an input A of τ -synthesis allows a positive decision, we want to constructa corresponding τ -net N . TS represents the behavior of a modeled system by means of global states (states of TS) and transitions between them (events). Dealing with a Petri net, we operate with local states (places) and their changing (transitions), while the global states of a net are markings, i.e., combinationsof local states. Since A and A N must be isomorphic, N ’s transitions correspond to A ’s events. Theconnection between global states in TS and local states in the sought net is given by regions of TS thatmimic places: A τ -region R = ( sup , sig ) of A = ( S , E , δ , ι ) consists of the support sup : S → { , } and the signature sig : E → E τ where every edge s e s (cid:48) of A leads to an edge sup ( s ) sig ( e ) sup ( s (cid:48) ) oftype τ . If P = q e . . . e n q n is a path in A , then P R = sup ( q ) sig ( e ) . . . sig ( e n ) sup ( q n ) is a path in τ . We say P R is the image of P (under R ). Notice that R is implicitly defined by sup ( ι ) and sig : Since A is reachable, for every state s ∈ S ( A ) , there is a path ι e . . . e n s n such that s = s n . Thus, since τ isdeterministic, we inductively obtain sup ( s i + ) by sup ( s i ) e i sup ( s i + ) for all i ∈ { , . . . , n − } and s = ι .Consequently, we can compute sup and, thus, R purely from sup ( ι ) and sig , cf. Figure 5 and Example 3.A region ( sup , sig ) models a place p and the associated part of the flow function f . In particular, f ( p , e ) = sig ( e ) and M ( p ) = sup ( s ) , for marking M ∈ RS ( N ) that corresponds to s ∈ S ( A ) . Every set R of τ -regions of A defines the synthesized τ -net N R A = ( R , E , f , M ) with f (( sup , sig ) , e ) = sig ( e ) and M (( sup , sig )) = sup ( ι ) for all ( sup , sig ) ∈ R , e ∈ E . State and Event Separation.
To ensure that the input behavior is captured by the synthesized net,we have to distinguish global states, and prevent the firings of transitions when their corresponding x nop ( x ) inp ( x ) out ( x ) set ( x ) res ( x ) swap ( x ) used ( x ) free ( x ) i of I . If a cell is empty, then i is undefined on the respective x . nop nopinp , swapswap τ R a inp b nop R swapinp N A N ( , ) ( , ) ( , ) a b Figure 2: The type τ = { nop , inp , swap } and a τ -net N and its reachability graph A N .2 Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency nopfree nopinp τ s s aa A r r bc A nop nopusedsetswapset , swap τ Figure 3: The type τ = { nop , inp , free } , the TSs A and A and the type τ = { nop , swap , used , set } . R a swap N A N (0) (1) aa Figure 4: The 1-restricted τ -net N , where τ is defined according to Figure 3 and N = N R A according toExample 2, and its reachability graph A N . A ι s s s a b c A R used swap set Figure 5: The TS A , a simple directed path and its image A R under R , with R from Example 3.events are not present in TS. This is stated as so called separation atoms and problems . A pair ( s , s (cid:48) ) ofdistinct states of A defines a states separation atom (SSP atom). A τ -region R = ( sup , sig ) solves ( s , s (cid:48) ) if sup ( s ) (cid:54) = sup ( s (cid:48) ) . If every SSP atom of A is τ -solvable then A has the τ -states separation property ( τ -SSP,for short). A pair ( e , s ) of event e ∈ E and state s ∈ S where e does not occur, that is ¬ s e , defines an event/state separation atom (ESSP atom). A τ -region R = ( sup , sig ) solves ( e , s ) if sig ( e ) is not definedon sup ( s ) in τ , that is, ¬ sup ( s ) sig ( e ) . If every ESSP atom of A is τ -solvable then A has the τ -event/stateseparation property ( τ -ESSP, for short). A set R of τ -regions of A is called τ - admissible if for each SSPand ESSP atom there is a τ -region R in R that solves it. We say that A is τ -solvable if it has a τ -admissibleset. The next lemma establishes the connection between the existence of τ -admissible sets of A and theexistence of a τ -net N that solves A : Lemma 1 ([3]) . A TS A is isomorphic to the reachability graph of a τ -net N if and only if there is a τ -admissible set R of A such that N = N R A . Example 2.
Let τ , τ , A and A be defined in accordance to Figure 3. The TS A has no ESSPatoms. Hence, it has the τ -ESSP and τ -ESSP. The only SSP atom of A is ( s , s ) . It is τ -solvable byR = ( sup , sig ) with sup ( s ) = , sup ( s ) = , sig ( a ) = swap . Thus, A has the τ -admissible set R = { R } , and the τ -net N = N R A = ( { R } , { a } , f , M ) with M ( R ) = sup ( s ) and f ( R , a ) = sig ( a ) solves A . Figure 4 depicts N (left) and its reachability graph A N (right). The SSP atom ( s , s ) is not τ -solvable, thus, neither is A . TS A has ESSP atoms ( b , r ) and ( c , r ) , which are both τ -unsolvable.The only SSP atom ( r , r ) in A can be solved by the τ -region R = ( sup , sig ) with sup ( r ) = ,sup ( r ) = , sig ( b ) = set , sig ( c ) = swap . Thus, A has the τ -SSP, but not the τ -ESSP. None of the(E)SSP atoms of A can be solved by any τ -region. Notice that the τ -region R maps two events to asignature different from nop . Thus, in case of d-restricted τ -synthesis, R would not be valid for d = . onny Tredup and Evgeny Erofeev Example 3.
Let A be defined in accordance to Figure 5 and τ according to Figure 3. It definessup ( ι ) = , sig ( a ) = used , sig ( b ) = swap and sig ( c ) = set implicitly a τ -region R = ( sup , sig ) of A asfollows: sup ( s ) = δ τ ( , used ) = , sup ( s ) = δ τ ( , swap ) = and sup ( s ) = δ τ ( , set ) = . The imageA R of A (under R) is depicted on the right hand side of Figure 5. One easily verifies that δ A ( s , e ) = s (cid:48) implies δ τ ( sup ( s ) , sig ( e )) = sup ( s (cid:48) ) , cf. Figure 3. By Lemma 1, every τ -admissible set R implies that N R A τ -solves A . In this paper, we investigate thecomplexity of synthesising a solving τ -net N whose dependency number d N does not exceed a naturalnumber d . Recall that if R is a set of A ’s regions, then R ’s regions model places of the synthesized net N R A .Thus, d N R A ≤ d if and only if R is d-restricted , that is, every region R = ( sup , sig ) of R is d-restricted : |{ e ∈ E | sig ( e ) (cid:54) = nop }| ≤ d . By Lemma 1, this implies that there is a d -restricted τ -net N if and only ifthere is a d -restricted τ -admissible set R . This finally leads to the following parameterized problem thatis the main subject of this paper: Dependency Restricted τ -Synthesis (DR τ S) Input: a finite, reachable TS A , a natural number d . Parameter: dDecide: whether there exists a d -restricted τ -admissible set R of A . d -Restricted τ -Synthesis For a start, we observe that, similar to (unrestricted) τ -synthesis [19], DR τ S is in NP. Moreover, thereis a trivial reduction from τ -synthesis to DR τ S: Since a τ -region can map at most all events of a TS A = ( S , E , δ , ι ) not to nop , A is τ -solvable if and only if A is τ -solvable by | E | -restricted τ -regions. Thus,if τ -synthesis is NP-complete, then DR τ S is also NP-complete.Let’s argue that DR τ S belongs to the complexity class XP. Let A = ( S , E , δ , ι ) be a TS, d ∈ N and let | A | be the maximum number of edges that A possibly has, that is, | A | = | S | | E | . A τ -region R = ( sup , sig ) is implicitly defined by sup ( ι ) and sig . We are interested in regions of A for which there is an i ∈ { , . . . , d } such that |{ e ∈ E | sig ( e ) (cid:54) = nop }| = i . For every event e ∈ E , we have at most | τ | − ≤ nop . Since sup ( ι ) ∈ { , } , we have to consider at most 2 · d · ∑ di = (cid:0) | E | i (cid:1) regions atall, which can be estimated by O ( d | A | d ) To check if the chosen signature actually implies regions of A and to solve the (E)SSP atoms of A , weneed to construct the regions explicitly, that is, we have to compute sup . To do so, we firstly compute aspanning tree A (cid:48) of A , which is doable in time O ( | A | ) by the algorithm of Tarjan [18] and needs to bedone only once. In A (cid:48) , there is exactly one path from ι to s for all s ∈ S , and A (cid:48) has | S | − sup ( ι ) and sig , it costs time at most O ( | A | ) to compute sup . The effort to computeall potentially interesting regions explicitly is thus at most O ( d | A | d + ) . After that, we check for any fixedpotential region if it is actually a well-defined region, that is, whether s e s (cid:48) implies sup ( s ) sig ( e ) sup ( s (cid:48) ) .For a fixed region, this is doable in time O ( | A | ) . Thus the effort to compute all interesting regions of A is O ( d | A | d + ) .For a fixed separation atom ( s , s (cid:48) ) or ( e , s ) we simply have to check if sup ( s ) (cid:54) = sup ( s (cid:48) ) or if δ τ ( sup ( s ) , sig ( e )) is not defined, respectively, which is doable in time O ( | A | ) . Since we have at most O ( | A | ) separation atoms and at most O ( d | A | d ) regions, the check for the (E)SSP is doable in time O ( d | A | d + ) . Finally, if we add up the effort to get all interesting regions and the effort to check whetherthese regions witness the (E)SSP of A , then we obtain that the effort of the problem is bounded by O ( d | A | d + ) .4 Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency
On the other hand, in this section, we argue that DR τ S is W [ ] -hard for a range of Boolean types. Thefollowing theorem presents the result through the enumeration of these types. Theorem 1.
Dependency d-Restricted τ -Synthesis is W [ ] -hard if1. τ ⊇ { nop , inp , set } or τ ⊇ { nop , out , res } ,2. τ = { nop , set , res } ∪ ω or τ = { nop , set , res , swap } ∪ ω and /0 (cid:54) = ω ⊆ { free , used } ,3. τ = { nop , set , swap } ∪ ω , τ = { nop , out , set , swap } ∪ ω , τ = { nop , res , swap } ∪ ω or τ = { nop , inp , res , swap } ∪ ω and /0 (cid:54) = ω ⊆ { free , used } ,4. τ = { nop , inp , res , swap } or τ = { nop , out , set , swap } , Notice that, by the discussion above, for the types of Theorem 1, NP-completeness of DR τ S followsby the NP-completeness of τ -synthesis [19, p. 3]. The proof of Theorem 1 bases on parameterizedreductions of the problem Hitting Set , which is known to be W [ ] -complete (see e.g. [9]). The problem Hitting Set is defined as follows:
Hitting Set (HS)
Input: a finite set U , a set M = { M , . . . , M m } of subsets of U with M i = { X i , . . . , X i mi } and i < · · · < i m i for all i ∈ { , . . . , m } , a natural number κ . Parameter: κ Decide: whether there is a set S ⊆ U such that | S | ≤ κ and S ∩ M i (cid:54) = /0 for every i ∈ { , . . . , m } . The General Reduction Idea . An input I = ( U , M , κ ) of HS, where M = { M , . . . , M m } , is reduced toan instance ( A τ I , d ) of DR τ S with TS A τ I and d = f ( κ ) , for some linear function f . For every i ∈ { , . . . , m } ,the TS A τ I has a directed labeled path P i = s i , . . . s i , i (cid:96) − s i , i (cid:96) . . . s i , i mi X i X i (cid:96) − X i (cid:96) X i (cid:96) + X i mi that represents the set M i = { X i , . . . , X i mi } and uses its elements as events. The TS A τ I is then composedin such a way that for some ESSP atom α of A τ I the following is satisfied: If R = ( sup , sig ) is a d -restricted τ -region that solves α , then sup ( s i , ) (cid:54) = sup ( s i , i mi ) for all i ∈ { , . . . , m } . Since the image P Ri of P i is a directed path in τ , by sup ( s i , ) (cid:54) = sup ( s i , i mi ) , there has to be an element X ∈ M i such that s X s (cid:48) ∈ P i implies sup ( s ) (cid:54) = sup ( s (cid:48) ) . That is, the image sig ( X ) of X causes a state change on P Ri in τ .In particular, this implies sig ( X ) (cid:54) = nop . The following visualisation of P Ri sketches the situation for aregion R = ( sup , sig ) , where sup ( s i , ) = · · · = sup ( s i , i (cid:96) − ) = sup ( s i , i (cid:96) ) = · · · = sup ( s i , i mi ) = sig ( X i (cid:96) ) = set and sig ( X i k ) = nop for all k ∈ { , . . . , m i } \ { (cid:96) } : P Ri = sup ( s i , ) . . . sup ( s i , i (cid:96) − ) sup ( s i , i (cid:96) ) . . . sup ( s i , i mi ) sig ( X i ) sig ( X i (cid:96) − ) sig ( X i (cid:96) ) sig ( X i (cid:96) + ) sig ( X i mi ) nop nop nop nopset It is simultaneously true for all paths P , . . . , P m representing the sets M , . . . , M m , that on each paththere is a (not necessarily unique) X satisfying sig ( X ) (cid:54) = nop . Moreover, the reduction ensures that |{ X ∈ U | sig ( X ) (cid:54) = nop }| ≤ κ . In other words, S = { X ∈ U | sig ( X ) (cid:54) = nop } defines a sought hitting setof I . Thus, if ( A τ I , d ) is a yes-instance of DR τ S, implying the solvability of α , then I = ( U , M , κ ) is ayes-instance of HS.Conversely, if I = ( U , M , κ ) is a yes-instance, then there is a fitting τ -region of A τ I that solves α . Thereduction ensures that the d -restricted τ -solvability of α implies that all (E)SSP atoms of A τ I are solvableby d -restricted τ -regions. Thus, ( A τ I , d ) is a yes-instance, too. onny Tredup and Evgeny Erofeev ⊥ t , t , t , t , t , t , w k X X z k ⊥ t , t , t , t , t , t , w k X X z k ⊥ t , t , t , t , t , t , w k X X z k ⊥ t , t , t , t , t , t , t , w k X X X z k ⊥ h h h h h w k z o k (cid:9) (cid:9) (cid:9) (cid:9) Figure 6: The TS A τ I , where τ ⊇ { nop , inp , set } and I originates from Example 4. The green colored areasketches the states that are mapped to 1 by the region R X , , solving ( X , s ) for all s ∈ {⊥ , t , , t , } .In the following, we present the corresponding reductions and show that the solvability of α impliesthe existence of a sought-for hitting set. Moreover, we argue that the existence of a sought set implies the τ -solvability of α and, finally, the τ -solvability of A τ I .As an instance, the following (running) example serves for all concrete reductions that we present, tosimplify the understanding of the reductions’ formal descriptions. Example 4.
The input I = ( U , M , κ ) is defined by U = { X , X , X , X } and M = { M , M , M , M } , whereM = { X , X } , M = { X , X } , M = { X , X } and M = { X , X , X } , and κ = . A fitting hitting set ofM is given by S = { X , X } . Theorem 1.1: The Reduction.
In accordance to our general approach, we first define d = κ +
2. Next,we introduce the TS A τ I . Figure 6 provides a concrete example of A τ I , where I corresponds to Example 4.The TS A τ I has the following gadget H that applies the events k , z and o and provides the atom α = ( k , h ) : ⊥ m + h h h h h w m + k z o k For all i ∈ { , . . . , m } , the TS A τ I has the following gadget T i that applies w i , k , z and the elements of M i = { X i , . . . , X i mi } as events: ⊥ i t i , t i , . . . t i , m i + t i , m i + t i , m i + w i k X i X i mi z k The TS A τ I has the events (cid:9) , . . . , (cid:9) m to connect the gadgets T , . . . , T m and H by ⊥ (cid:9) . . . (cid:9) m ⊥ m + .The initial state of A τ I is ⊥ . Theorem 1.1: The Solvability of α Implies a Hitting Set.
We argue for τ ⊇ { nop , inp , set } , thehardness of the other types follows by symmetry. In the following, we argue that if there is a d -restricted τ -region R = ( sup , sig ) that solves α , then I has a hitting set of size at most κ . Let R = ( sup , sig ) be sucha τ -region. Since R solves α , we have either sig ( k ) ∈ { inp , used } and sup ( h ) = sig ( k ) ∈ { out , free } and sup ( h ) =
1. In what follows, we consider to the former case. The proof for the latter case issymmetrical.6
Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency If sig ( k ) = inp and sup ( h ) =
0, then s k s (cid:48) implies sup ( s ) = sup ( s (cid:48) ) =
0. By sup ( h ) = sup ( h ) =
1, we get sig ( o ) ∈ { out , set , swap } . In particular, since R is d -restricted, there are atmost κ events left that have a signature different from nop . By sup ( h ) = sup ( h ) = h z h ,we have sig ( z ) ∈ { nop , res , free } . Moreover, by sup ( t i , m i + ) = z t i , m i + , we have sig ( z ) = nop .By sig ( k ) = inp and sig ( z ) = nop , we conclude sup ( t i , ) = sup ( t i , m i + ) = i ∈ { , . . . , m } .Consequently, for every i ∈ { , . . . , m } , there is X ∈ M i such that sig ( X ) ∈ { out , set , swap } . Otherwise astate change from 0 to 1 would not be possible. Since R is d -restricted and sig ( k ) (cid:54) = nop (cid:54) = sig ( o ) , we get |{ X ∈ U | sig ( X ) (cid:54) = nop }| ≤ κ . This implies that S = { X ∈ U | sig ( X ) (cid:54) = nop } is a fitting hitting set of I .If sig ( k ) = used and sup ( h ) =
0, then s k s (cid:48) implies sup ( s ) = sup ( s (cid:48) ) =
1. By sup ( h ) = sup ( h ) = sup ( h ) =
0, we get sig ( z ) ∈ { inp , res , swap } and sig ( o ) ∈ { out , set , swap } . By sup ( t i , m i + ) = z t i , m i + , we get sig ( z ) = swap . Since R is d -restricted, there are at most κ − nop . Moreover, by sig ( k ) = used and sig ( z ) = swap , we have sup ( t i , ) = sup ( t i , m i + ) = i ∈ { , . . . , m } . Just like before, we conclude that S = { X ∈ U | sig ( X ) (cid:54) = nop } is asought hitting set of I .Conversely, a κ -HS of ( U , M , κ ) implies the τ -solvability of A τ I , which is the statement of the followinglemma. Due to space restrictions, we omit the proof which can be found in [23]. Lemma 2.
Let τ be a type of nets in correspondence of Theorem 1.1. If ( U , M , κ ) has a κ -HS, then thereis a d-restricted admissible set of A I τ . Theorem 1.2: The Reduction.
Let τ be a type of Theorem 1.2. According to our general approach,we first define d = κ +
4. Next we introduce the TS A τ I . Figure 7 provides an example of A τ I , where I corresponds to Example 4. The TS A τ I has the following gadget H that provides the atom α = ( k , h , ) : ⊥ m + h , h , h , h , h , w m + k o o kw m + k o o k Moreover, the TS A τ I has the following gadgets H and H : H = ⊥ m + h , h , h , w m + k z , o w m + k z H = ⊥ m + h , w m + , o , z w m + For all i ∈ { , . . . , m } , TS A τ I has the following gadget T i that applies w i , k , z , z and the elements of M i = { X i , . . . , X i mi } as events: ⊥ i t i , t i , t i , . . . t i , m i + t i , m i + t i , m i + w i k z X i X i mi z kw i k z X i X i mi z k Finally, the TS A τ I uses the events (cid:9) , . . . , (cid:9) m + and applies for all i ∈ { , . . . , m } the edges ⊥ i (cid:9) i ⊥ i + and ⊥ i + (cid:9) i ⊥ i + to join the gadgets T , . . . , T m and H , H , H . Theorem 1.2: The τ -solvability of α Implies a Hitting Set.
Let R = ( sup , sig ) be a τ -region thatsolves α , that is, either sig ( k ) = used and sup ( h , ) = sig ( k ) = free and sup ( h , ) =
1. In thefollowing, we assume that sig ( k ) = used and sup ( h , ) =
0. The arguments for the case sig ( k ) = free onny Tredup and Evgeny Erofeev ⊥ t , t , t , t , t , t , t , w k z X X z kw k z X X z k ⊥ t , t , t , t , t , t , t , w k z X X z kw k z X X z k ⊥ t , t , t , t , t , t , t , w k z X X z kw k z X X z k ⊥ t , t , t , t , t , t , t , t , w k z X X X z kw k z X X X z k ⊥ h , h , h , h , h , w k o o kw k o o k ⊥ h , h , h , w k z , o w k z ⊥ h , w , o , z w (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) Figure 7: The TS A τ I where τ corresponds to Theorem 1.2 and I to Example 4 with the HS S = { X , X } .The green colored area sketches the τ -region R = ( sup , sig ) that solves α , where, for all e ∈ E ( A τ I ) ,if e = k , then sig ( e ) = used ; if e ∈ { o } ∪ S , then sig ( e ) = set ; if e ∈ { o , z , (cid:9) } , then sig ( e ) = res ;otherwise sig ( e ) = nop .and sup ( h , ) = s e s (cid:48) ∈ A τ I , then s (cid:48) e s (cid:48) ∈ A τ I . Thus, for all e ∈ E ( A τ I ) holds sig ( e ) (cid:54) = swap .Since sig ( k ) = used , if s k s (cid:48) , then sup ( s ) = sup ( s (cid:48) ) =
1. In particular, we have sup ( t i , m i + ) = i ∈ { , . . . , m } . Moreover, by sup ( h , ) = sup ( h , ) =
0, we have sig ( o ) = res and sig ( o ) = set .This implies sup ( h , ) = sup ( h , ) =
0. By sup ( h , ) = sup ( h , ) =
0, we get sig ( z ) = res ; by sup ( h , ) = sup ( t , m i + ) =
1, we get sig ( z ) = nop . Thus, by sig ( z ) = res and sig ( z ) = nop , weget sup ( t i , ) = sup ( t i , m i + ) = i ∈ { , . . . , m } . Consequently, for all i ∈ { , . . . , m } , there is X ∈ M i such that sig ( X ) = set . Since sig ( e ) (cid:54) = nop for all e ∈ { k , o , o , z } and R is d -restricted, it holds |{ X ∈ U | sig ( X ) (cid:54) = nop }| ≤ κ . This implies that S = { X ∈ U | sig ( X ) (cid:54) = nop } is a sought-for hitting setof I .In return, if ( U , M , κ ) has a κ -HS, then A I τ is τ -solvable, which is the statement of the followinglemma. Due to space restrictions, we omit the proof which can be found in [23]. Lemma 3.
Let τ be a type of nets in correspondence of Theorem 1.2. If ( U , M , κ ) has a κ -HS, then thereis a d-restricted admissible set of A I τ . Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency
Theorem 1.3: The Reduction.
We restrict ourselves to the case where τ = { nop , set , swap } ∪ ω or τ = { nop , out , set , swap } ∪ ω and /0 (cid:54) = ω ⊆ { free , used } . The hardness for the other types follows bysymmetry. First, we define d = κ +
4. Next, we introduce the TS A τ I . Figure 8 provides a full example of A τ I where I corresponds to Example 4.The TS A τ I has the following gadgets H and H that provide the atom α = ( k , h , ) : H = ⊥ m + h , h , h , h , h , w m + k o o k H = ⊥ m + h , h , h , h , h , h , w m + k z o z k Moreover, for every i ∈ { , . . . , m } , the TS A τ I has the following gadget T i that has the elements of M i = { X i , . . . , X i mi } as events: ⊥ i t i , t i , t i , t i , t i , t i , t i , ... t i , m i − t i , m i − t i , m i t i , m i + t i , m i + t i , m i + t i , m i + w i k z a i , X i X i a i , a i , m i X i mi X i mi a i , m i z k Notice that, for all (cid:96) ∈ { , . . . , m i } , the event a i ,(cid:96) that encompasses the event X i (cid:96) of M i is bounded to theoccurrence of X i (cid:96) in T i . In particular, if two distinct sets M i and M j share an event X ∈ U , that is, there areindices (cid:96) ∈ { , . . . , m i } and n ∈ { , . . . , m j } such that X = X i (cid:96) = X j n , then a i ,(cid:96) embraces X in T i and a j , n embraces X in T j but a i ,(cid:96) and a j , n are distinct. Finally, to obtain A τ I , we use fresh events (cid:9) , . . . , (cid:9) m + and connect T , . . . , T m , H and H by ⊥ (cid:9) . . . (cid:9) m + ⊥ m + . The initial state of A τ I is ⊥ . Notice thatfor every region R of A τ I , holds that s e s (cid:48) ∈ A τ I and sup ( s ) (cid:54) = sup ( s (cid:48) ) implies sig ( e ) = swap . Moreover,if s e s (cid:48) ∈ A τ I , then, by construction, s (cid:48) e . By the definition of out , this implies sig ( e ) (cid:54) = out for all e ∈ E ( A τ I ) . Theorem 1.3: The τ -Solvability of α Implies a Hitting Set.
Let R = ( sup , sig ) be a τ -regionthat solves α . Since R solves α , we have either sig ( k ) = used and sup ( h , ) = sig ( k ) = free and sup ( h , ) =
1. In the following, we consider the former case, the arguments for the latter aresymmetrical. Please note Figure 8 during the following considerations. By sig ( k ) = used , we havethat sup ( s ) = sup ( s (cid:48) ) = s k s (cid:48) ∈ A τ I . In particular, we have sup ( h , ) = sup ( h , ) = sup ( h , ) =
0, implies sig ( o ) = sig ( o ) = swap . Moreover, we have sup ( h , ) = sup ( h , ) = P R of the path P = h , z . . . z h , is even.Since sig ( o ) = swap , this implies that there is exactly one event e ∈ { z , z } such that sig ( e ) = swap . Weconsider the case sig ( z ) = swap . The arguments for the case sig ( z ) = swap are similar. The region R is d -restricted, and k , o , o , z have signatures different from nop . There are at most κ events left whosesignatures are not nop .Let i ∈ { , . . . , m } be arbitrary but fixed. By sig ( k ) = used , we have sup ( t i , ) = sup ( t i , m i + ) =
1. By sig ( z ) = swap and sig ( z ) (cid:54) = swap , this implies sup ( t i , ) = sup ( t i , m i + ) =
1. Hence the image P R of the path P = t i , t i , t i , t i , t i , . . . t i , m i − t i , m i − t i , m i t i , m i + t i , m i + a i , X i X i a i , a i , m i X i mi X i mi a i , m i is a path from 0 to 1 in τ . Thus, there is an event e ∈ { X i , . . . , X i mi }∪{ a i , , . . . , a i , m i } whose signature causesthe state change from 0 to 1. This implies sig ( e ) (cid:54) = nop . Assume, for a contradiction, that sig ( e ) = nop for onny Tredup and Evgeny Erofeev ⊥ t , t , t , t , t , t , t , t , t , t , t , t , t , w k z a , X X a , a , X X a , z k ⊥ t , t , t , t , t , t , t , t , t , t , t , t , t , w k z a , X X a , a , X X a , z k ⊥ t , t , t , t , t , t , t , t , t , t , t , t , t , w k z a , X X a , a , X X a , z k ⊥ t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , w k z a , X X a , a , X X a , a , X X a , z kh , h , h , h , h , ⊥ w k o o kh , h , h , h , h , h , ⊥ w k z o z k (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) Figure 8: A full example of A τ I , where τ belongs to the types of Theorem 1.3 and I originates fromExample 4. Green colored area: A sketch of the { nop , set , swap , used } -region R k = ( sup , sig ) , based onthe HS S = { X , X } , that satisfies sig ( k ) = used and sup ( h , ) = α .all e ∈ { X i , . . . , X i mi } . Let (cid:96) ∈ { , . . . , m i } be arbitrary but fixed. By sig ( X (cid:96) ) = nop , we get sup ( t i , (cid:96) − ) = sup ( t i , (cid:96) ) = sup ( t i , (cid:96) + ) . Recall that sup ( s ) (cid:54) = sup ( s (cid:48) ) implies sig ( e ) = swap for all s e s (cid:48) ∈ A τ I . Thus, if sig ( a i ,(cid:96) ) (cid:54) = swap , then sup ( t i , (cid:96) − ) = sup ( t i , (cid:96) − ) = sup ( t i , (cid:96) ) = sup ( t i , (cid:96) + ) = sup ( t i , (cid:96) + ) . Otherwise, if sig ( a i ,(cid:96) ) = swap , then sup ( t i , (cid:96) − ) (cid:54) = sup ( t i , (cid:96) − ) = sup ( t i , (cid:96) ) = sup ( t i , (cid:96) + ) (cid:54) = sup ( t i , (cid:96) + ) . Consequently,both cases imply sup ( t i , (cid:96) − ) = sup ( t i , (cid:96) + ) . Since (cid:96) was arbitrary, this implies sup ( t i , ) = sup ( t i , m i + ) , acontradiction. Hence, there is an event e ∈ { X i , . . . , X i mi } such that sig ( e ) (cid:54) = nop . Since i was arbitrary,this is simultaneously true for all T , . . . , T m . Moreover, since R respects the parameter, the cardinality of S = { X ∈ U | sig ( X ) (cid:54) = nop } is at most κ . Thus, S is a fitting hitting set of I .The next lemma completes the proof of Theorem 1.3 and states that a sought HS of I implies a d -restricted admissible set of A τ I . Due to space restrictions, its proof can be found in [23]. Lemma 4.
Let τ be a type of net corresponding to Theorem 1.3. If I = ( U , M , κ ) has a fitting HS, then A τ I has a d-restricted admissible set. Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency
Theorem 1.4: The Reduction
In the following, we argue for τ = { nop , inp , res , swap } . The hardness for τ = { nop , out , set , swap } then follows by symmetry. For a start, we define d = κ +
4. The TS A τ I has thefollowing gadgets H , . . . , H that provide the atom α = ( k , h , ) : H = ⊥ m + h , h , h , h , h , w m + k o o k H = ⊥ m + h , h , h , h , h , w m + k z o k H = ⊥ m + h , h , h , h , h , w m + k z o k H = ⊥ m + h , h , h , h , h , h , w m + k z z z k H = ⊥ m + h , h , h , h , h , h , w m + k z z z k Moreover, for every i ∈ { , . . . , m } , the TS A τ I has the following gadget T i that uses the elements of M i = { X i , . . . , X i mi } as events: t i , t i , t i , . . . t i , m i + t i , m i + t i , m i + k z X i X i mi z k The Joining of A τ I by Relevant Paths. Similar to the previous reductions, we essentially want to connectall gadgets by a simple directed path on which every event occurs exactly once. However, since we want toensure that if α is τ -solvable then all (E)SSP atoms of A τ I are also τ -solvable (by d -restricted regions), thisis not directly possible for the gadgets T , . . . , T m . Instead, we complete the construction of A τ I through twofurther steps. Firstly, for all i ∈ { , . . . , m } , we extend the gadget T i to a (path-) gadget G i = ⊥ i T i with starting state ⊥ i . Secondly, we use the events (cid:9) , . . . , (cid:9) m + and connect the gadgets G , . . . , G m and H , . . . , H by ⊥ (cid:9) ⊥ (cid:9) . . . (cid:9) m + ⊥ m + . The resulting TS is A τ I , and its initial state is ⊥ . Before weintroduce the definition of G i , in the following, we briefly outline which obstacles arise and, in order toovercome them, in which way they lead to G i .Let i ∈ { , . . . , m } and (cid:96) ∈ { , . . . , m i } be arbitrary but fixed. Similar to the approach of region R X , i ,(cid:96) ofTheorem 1.1, which is sketched for i = (cid:96) = X i (cid:96) “gadget-wise”.In particular, to solve ( X i (cid:96) , s ) for all predecessor states s of t i ,(cid:96) + in G i , that is, ⊥ i , . . . , t i ,(cid:96) , we want toconstruct a region R = ( sup , sig ) such that as few events as possible are not mapped to nop . (Independentof A τ I ’s size, the region R , Xi ,(cid:96) of Theorem 1.1 maps four events not to nop .) First of all, look at the followingdefinition: sup ( ⊥ ) =
0; for all e ∈ E ( A τ I ) , if e = X i (cid:96) , then sig ( e ) = inp ; if e is X i (cid:96) ’s direct predecessor,that is, e t i ,(cid:96) + , then sig ( e ) = swap ; otherwise sig ( e ) = nop . In Figure 9, the red colored area sketchesthis region for X = X and its direct predecessor z ; the green colored area sketches this region for X = X and its direct predecessor X . Actually, R is always well defined if X i (cid:96) ∈ E ( T j ) implies that X i (cid:96) ’s direct predecessor e t i ,(cid:96) + also belongs to E ( T j ) . This is not true if there is an occurrence of X i (cid:96) in a gadget T j , say at t j ,(cid:96) (cid:48) , such that X i (cid:96) ’s predecessor does not belong to T j ’s event set. For example,consider in Figure 9 the event X = X of T that occurs as X in T . In T , X is directly preceded by X , but X does not occur in T . The following problem arises. Since sig ( X i (cid:96) ) = inp , there has to bean event e on the unambiguous path ⊥ . . . t j ,(cid:96) (cid:48) such that sig ( e ) = swap . Otherwise, X i (cid:96) ’s source t i ,(cid:96) (cid:48) in T j would not satisfy sup ( t i ,(cid:96) (cid:48) ) =
1. At first glance, a possible solution might be to implement anadditional (unique) event y j on the path ⊥ j t j , for all j ∈ { , . . . , m } where X i (cid:96) belongs to E ( T j ) onny Tredup and Evgeny Erofeev ⊥ t , t , t , t , t , t , t , k z X X z k ⊥ t , t , t , t , t , t , t , k z X X z k ⊥ t , t , t , t , t , t , t , k z X X z k ⊥ t , t , t , t , t , t , t , t , k z X X X z k ⊥ ⊥ ⊥ ⊥ ⊥ ... ... ... ... ... (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) Figure 9: A snippet of A τ I ( τ = { nop , inp , res , swap } ) built from Example 4 and showing the gadgets T , . . . , T . Red colored area: the region R = ( sup , sig ) where sup ( ⊥ ) = sig ( X ) = inp ; sig ( z ) = swap ; sig ( e ) = nop for all e ∈ E ( A τ I ) \{ z , X } . Green colored area: the region R = ( sup , sig ) where sup ( ⊥ ) = sig ( X ) = inp ; sig ( X ) = swap ; sig ( e ) = nop for all e ∈ E ( A τ I ) \ { X , X } . ⊥ t , t , t , t , t , t , t , k z X X z k ⊥ t , t , t , t , t , t , t , y k z X X z k ⊥ t , t , t , t , t , t , t , k z X X z k ⊥ t , t , t , t , t , t , t , t , k z X X X z k (cid:9) (cid:9) (cid:9) Figure 10: A sketch of the “first-glance” solution for A τ I ( τ = { nop , inp , res , swap } ), where I correspondsto Example 4. Green colored area: the region R = ( sup , sig ) where sup ( ⊥ ) = sig ( X ) = inp ; sig ( X ) = sig ( y ) = swap ; sig ( e ) = nop for all e ∈ E ( A τ I ) \ { X , X , y } . s i , ji , s i , ji , s i , ji , v i , j ⊕ i , j s i , ji , s i , ji , s i , ji , s i , ji , v i , j ⊕ i , j ⊕ i , j s i , ji , s i , ji , s i , ji , s i , ji , s i , ji , v i , j ⊕ i , j ⊕ i , j ⊕ i , j ... s i , ji (cid:96) , s i , ji (cid:96) , s i , ji (cid:96) , ... s i , ji (cid:96) ,(cid:96) − s i , ji (cid:96) ,(cid:96) − s i , ji (cid:96) ,(cid:96) s i , ji (cid:96) ,(cid:96) + v i , j (cid:96) ⊕ i , j (cid:96) ⊕ i , j (cid:96) − ⊕ i , j ⊕ i , j ⊕ i , j ⊕ i , j Figure 11: The pyramidal approach of the relevant paths ensures that ⊕ -events are solvable by regionsindependent of the size of ( U , M , κ ) . Green colored area: a region R = ( sup , sig ) solving ( ⊕ i , j , s ) for allrelevant s ∈ S ( A τ I ) : sup ( ⊥ ) =
0; for all e ∈ E ( A τ I ) , if e = ⊕ i , j , then sig ( e ) = inp ; if e ∈ { v i , j , ⊕ i , j } , then sig ( e ) = swap ; otherwise sig ( e ) = nop . Blue colored area: a corresponding region solving ⊕ i , j . Theseregions are independent from the positions of G i , . . . , G i (cid:96) in A τ I or P i n in G i n , where n ∈ { , . . . , (cid:96) } .2 Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency but X i (cid:96) ’s direct predecessor event does not. Then we would modify the region R = ( sup , sig ) in a way, that sig ( y j ) = swap for all relevant j . Figure 10 sketches the situation for y .Unfortunately, for this construction and the sketched region, |{ e ∈ E ( A τ I ) | sig ( e ) (cid:54) = nop }| ≥ n + n is the number of gadgets in which X i (cid:96) occurs but its predecessor does not. Since X i (cid:96) couldoccur in numerous sets, in general, n depends on the size of M and does not necessarily respect theparameter d . Thus, this approach yields not a parameterized reduction. The next inelaborate solution toovercome this obstacle is to ensure that there is the same event, say y , on every path ⊥ j t j , for all j ∈ { , . . . , m } \ { i } such that X i (cid:96) ∈ E ( T j ) but X i (cid:96) ’s predecessor is not in E ( T j ) . However, one has to ensurethat the already discussed difficulties are not transferred from X i (cid:96) to y . Our solution uses relevant paths torealize a pyramidal approach that is sketched by Figure 11. Instead of one single event y (whose role isplayed by ⊕ i , j in Figure 11), this approach implements for every corresponding T j a unique directed path.Let i ∈ { , . . . , m } be arbitrary but fixed. We extend the gadget T i to G i = ⊥ i w i P i u i T i with startingstate ⊥ i and events w i , u i that embrace the path P i , to be defined next. To be able to refer uniformly tothe events X i , . . . , X i mi and z , we define e i = X i , . . . , e im i = X i mi and e im i + = z . Let j ∈ { , . . . , m i + } be arbitrary but fixed and let i < · · · < i (cid:96) ∈ { , . . . , m } \ { i } be exactly the indices different from i suchthat for the gadgets T i , . . . , T i (cid:96) we have e ij ∈ E ( T i n ) and e ij − (cid:54)∈ E ( T i n ) , for all n ∈ { , . . . , (cid:96) } . For all n ∈ { , . . . , (cid:96) } , we say that e ij is relevant for G i n and P i , ji n , n = s i , ji n , v i , jn s i , ji n , ⊕ i , jn s i , ji n , ⊕ i , jn − . . . ⊕ i , j s i , ji n , n + is the relevant path of G i n that originates from e ij . Example 5.
The event e = z of T of Figure 9 is preceded by e = X . While the event z occurs in T , T and T , the event X occurs in T but not in T and not in T . Thus, e is (only) relevant for T = T i andT = T i , where i = and i = . The corresponding relevant paths areP , , = s , , v , s , , ⊕ , s , , and P , , = s , , v , s , , ⊕ , s , , ⊕ , s , , . Equipped with these definitions, we are prepared to define the gadget G i . If there are no relevant eventsfor G i , then G i = ⊥ i w i q i u i T i . In particular, P i = q i . Otherwise, let e i j , . . . , e i n j n be the events that arerelevant for G i where i ≤ i ≤ · · · ≤ i n and j ≤ j ≤ · · · ≤ j n . Let P i , j i ,(cid:96) , P i , j i ,(cid:96) , . . . , P i n , j n i ,(cid:96) n be the relevantpaths of G i that origin from e i j , . . . , e i n j n , respectively. The path P i then originates from G i ’s relevant paths: G i = ⊥ i w i P i , j i ,(cid:96) c i P i , j i ,(cid:96) c i . . . c in P i n , j n i ,(cid:96) n u i T i See [23] for a full example.
Theorem 1.4: The τ -Solvability of α Implies a Hitting Set.
Let R = ( sup , sig ) be a d -restricted τ -region of A τ I that solves α . Since R solves α , one easily finds that sig ( k ) = inp and sup ( h , ) = sig ( k ) = inp , we have sup ( h , ) =
1; and sup ( h , ) = sig ( o ) = swap . Moreover, by sig ( k ) = inp and sig ( o ) = swap , we obtain that sup ( h , ) = sup ( h , ) = sup ( h , ) = sup ( h , ) =
0. Thisimplies sig ( z ) , sig ( z ) ∈ { nop , res } . By sig ( k ) = inp and sig ( z ) , sig ( z ) ∈ { nop , res } , we get sup ( h , ) = sup ( h , ) = sup ( h , ) = sup ( h , ) =
1. This implies sig ( z ) = sig ( z ) = swap . Since d = κ + R is d -restricted, there are at most κ events left whose signature is different from nop . Let i ∈ { , . . . , m } be arbitrary but fixed. By sig ( k ) = inp , we get sup ( t i , ) = sup ( t i , m i + ) =
1. Moreover, by sig ( z ) = sig ( z ) = swap , we get sup ( t i , ) = sup ( t i , m i + ) =
0. Thus, there is an event X ∈ E ( T i ) such that onny Tredup and Evgeny Erofeev sig ( X ) ∈ { inp , res , swap } . Since i was arbitrary and R is d -restricted, the set S = { X ∈ U | sig ( X ) (cid:54) = nop } is a sought-for HS of I . Theorem 1.4: A Hitting Set Implies the τ -Solvability of A τ I . We argue for the τ -solvability of k , implying the τ -solvability of α . The following d -restricted τ -region R = ( sup , sig ) solves α andsolves ( k , s ) for all relevant s ∈ (cid:83) mi = S ( H i ) \ {⊥ m + , . . . , ⊥ m + } , too: sup ( ⊥ ) =
1; for all e ∈ E ( A τ I ) , if e = k , then sig ( e ) = inp ; if e ∈ { o , z , z } , then sig ( e ) = swap ; if e ∈ S , then sig ( e ) = res ; otherwise, sig ( e ) = nop .Let i ∈ { , . . . , m } be arbitrary but fixed. The following region R = ( sup , sig ) solves ( k , s ) for allrelevant s ∈ S ( G i ) : If i =
1, then sup ( ⊥ ) =
0, otherwise sup ( ⊥ ) =
1; for all e ∈ E ( A τ I ) , if e ∈ { k , (cid:9) i − } ,then sig ( k ) = inp ; if e ∈ {(cid:9) i , o , z , z , z } , then sig ( e ) = swap ; if e = z , then sig ( e ) = res ; otherwise, sig ( e ) = nop . It is easy to see that, for any s ∈ {⊥ m + , . . . , ⊥ m + } , this region can be modified to a d -restricted region that solves ( k , s ) .Let i ∈ { , . . . , m i } be arbitrary but fixed. The separability of X i , . . . , X i mi , z in G i has already beensketched in the explanation of the relevant paths. Clearly, these events are separable in the gadgets inwhich they do not occur. Also the helper events of the relevant paths are separable. We omit the proofs forthe sake of readability. In this paper, we investigate the parameterized complexity of DR τ S parameterized by d and show W [ ] -hardness for a range of Boolean types. As a result, d is ruled out for fpt-approaches for the considered typesof nets. As future work, it remains to classify DR τ S exactly in the W -hierarchy. Moreover, one may lookfor other more promising parameters: If N = ( P , T , M , f ) is a Boolean net, p ∈ P and if the occupationnumber o p of p is defined by o p = |{ M ∈ RS ( N ) | M ( p ) = }| then the occupation number o N of N isdefined by o N = max { o p | p ∈ P } . If R is a τ -admissible set (of a TS A ) and R ∈ R , then the supportof R determines the number of markings of N R A that occupy R , that, is, o R = |{ s ∈ S ( A ) | sup ( s ) = }| .Thus, searching for a τ -net where o N ≤ n , n ∈ N , corresponds to searching for a τ -admissible set R such that |{ s ∈ S ( A ) | sup ( s ) = }| ≤ n for all R ∈ R . As a result, for each (E)SSP atom α there are atmost O ( (cid:0) | S | o N (cid:1) ) fitting supports for τ -regions solving α . Thus, the corresponding problem o N -restricted τ -synthesis parameterized by o N is in XP if, in a certain sense, τ -regions are fully determined by a givensupport sup . References [1] Wil M. P. van der Aalst (2011):
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